A voltage divider sets the inputs of the two op-amps (used as comparators).
The upper op-amp compares the trigger input to 1/3 the supply voltage. The
lower op-amp compares the threshold input to 2/3 the supply voltage.

A timing interval starts when the trigger input goes low enough to trigger
the upper op-amp. That sets the
flip-flop, causing the output
to go high. The 555 waits for the threshold input to trigger the lower
op-amp. As the capacitor charges, the threshold
input slowly rises until it reaches the required level. Then, the op-amp
resets the flip-flop, bringing the output low. The flip-flop's inverted
output also provides current to the base of the transistor on the bottom,
which discharges the capacitor through the discharge input.

When the capacitor is discharged enough so that the upper op-amp is
triggered again, a new timing interval begins.